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I would like to know if defining a simple closed shape as a set of points in a plane (for example, a polygon or a disc) is technically correct in terms of math. What do I mean: in school, I learned from geometry textbooks that any shape is an infinite set of certain points on a plane. I have nothing against the definition of a curve (line) as an infinite set of points (because axiomatically neither a point nor a line has an area), but regarding shapes I stumble upon the paradox stating that an infinite set of zero-area points will form a certain region in the plane with a finite non-zero area.

Any disc (or polygon) does contain an infinite set of points, but isn't it supposed to be only a consequence, not a definition?

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    $\begingroup$ The paradox you mention can be cleared up with some measure theory. Simply put, any 'shape' is indeed a set of points in the plane. Like you mentioned, a line is a set of infinitely many points and yet has area zero. However, there is no problem in saying that a polygon, which is also made up of infinitely many points, has positive finite area. This goes to show that the definition of area does not take into account the points making up the shape, but instead it is defined in terms of a measure function which looks at the structure of the shape itself. $\endgroup$
    – M_B
    Commented Aug 26, 2017 at 20:50
  • $\begingroup$ It's a very good question. I guess for most people, a shape is what it is. The set theoretical definition is something like a small print, a legalese, to keep us on firm ground if trouble arise. I doubt even seasoned mathematicians think of shapes exclusively as "sets of points". $\endgroup$
    – orangeskid
    Commented Aug 26, 2017 at 22:21

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What you're missing is that a shape isn't a set of points — it is a set of points in context.

A much simpler example of this idea is that if you were interested in describing lines, you could do so with a set of only two points!

When you describe a line with just two points it's obvious that the set of points doesn't encode everything about the line. The set of all points doesn't encode everything about the line, it's just less obvious what's missing.

If you want to consider the line in isolation, you need more than just its set of points to faithfully describe it — for example, a topology or a metric or a measure or other things depending on what features you want to remember.

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